The average age of father and his two sons is 27 years. Five years ago of the two sons was 12 years. If the difference between the ages of two sons is 4 years,then the present age of the father is

Options

A. 34 years
B. 47 years
C. 64 years
D. 27 years

Answer

To find the present age of the father given the average age of the father and his two sons, we can follow these steps:

### Step 1: Understand the average age
The average age of the father and his two sons is given as 27 years. This means:
\[ \text{Average} = \frac{\text{Sum of ages}}{\text{Number of persons}} \]
Let the ages of the father and his two sons be \(x\), \(y\), and \(z\) respectively. Therefore, we can write:
\[ \frac{x + y + z}{3} = 27 \]
Multiplying both sides by 3 gives:
\[ x + y + z = 81 \quad \text{(Equation 1)} \]

### Step 2: Use the information about their ages 5 years ago
Five years ago, the sum of the ages of the two sons was 12 years. This can be expressed as:
\[ (y - 5) + (z - 5) = 12 \]
Simplifying this gives:
\[ y + z - 10 = 12 \]
Adding 10 to both sides results in:
\[ y + z = 22 \quad \text{(Equation 2)} \]

### Step 3: Use the difference in ages
We are also given that the difference between the ages of the two sons is 4 years. We can express this as:
\[ |y - z| = 4 \]
Assuming \(y > z\), we can write:
\[ y - z = 4 \quad \text{(Equation 3)} \]

### Step 4: Solve the equations
Now we have two equations (Equation 2 and Equation 3):
1. \(y + z = 22\)
2. \(y - z = 4\)

We can solve these two equations simultaneously. Adding both equations gives:
\[ (y + z) + (y - z) = 22 + 4 \]
This simplifies to:
\[ 2y = 26 \implies y = 13 \]

Now, substituting \(y = 13\) back into Equation 2:
\[ 13 + z = 22 \implies z = 9 \]

### Step 5: Find the father's age
Now we have the ages of the two sons: \(y = 13\) and \(z = 9\). We can substitute these values back into Equation 1 to find the father's age:
\[ x + 13 + 9 = 81 \]
This simplifies to:
\[ x + 22 = 81 \implies x = 81 - 22 = 59 \]

Thus, the present age of the father is:
\[ \text{Father's age} = 59 \text{ years} \]

### Summary
The present age of the father is **59 years**.